Notes

Problems

Find the derivative of each function

\(\textbf{1)}\) \(f(x)=3x^4-2x+8\)

Show Derivative

The answer is \(12x^3-2\)

Show Work

\(\,\,\,\,\,\text{To differentiate } f(x) = 3x^4 – 2x + 8, \text{ apply the power rule to each term.}\)
\(\,\,\,\,\,f'(x) = 3 \cdot 4x^{3} – 2 + 0\)
\(\,\,\,\,\,f'(x) = 12x^3 – 2\)

 

\(\textbf{2)}\) \( f(x)=5-\frac{1}{x} \)

Show Answer

The derivative is \( f’ (x)=\displaystyle\frac{1}{x^{2}} \)

Show Work

\(\,\,\,\,\,\text{To differentiate } f(x) = 5 – \frac{1}{x}, \text{ rewrite } \frac{1}{x} \text{ as } x^{-1} \text{ and use the power rule.}\)
\(\,\,\,\,\,f(x) = 5 – x^{-1}\)
\(\,\,\,\,\,f'(x) = 0 + (-1)x^{-2}\)
\(\,\,\,\,\,f'(x) = \frac{1}{x^{2}}\)

 

\(\textbf{3)}\) \( f(x)=\frac{1}{2} x^{3}+4x^{2} \)

Show Answer

The derivative is \( f'(x)=\displaystyle\frac{3x^{2}}{2}+8x \)

Show Work

\(\,\,\,\,\,\text{Differentiate } f(x) = \frac{1}{2} x^3 + 4x^2 \text{ by applying the power rule to each term.}\)
\(\,\,\,\,\,f'(x) = \frac{1}{2} \cdot 3x^{2} + 4 \cdot 2x\)
\(\,\,\,\,\,f'(x) = \frac{3x^{2}}{2} + 8x\)

 

\(\textbf{4)}\) \( f(x)=\sin x-\cos x+2 \)

Show Answer

The derivative is \( f'(x)=\cos x+\sin x \)

Show Work

\(\,\,\,\,\,\text{Differentiate } f(x) = \sin x – \cos x + 2 \text{ by applying the derivatives of } \sin x, \cos x, \text{ and constants.}\)
\(\,\,\,\,\,f'(x) = \cos x + \sin x + 0 \)
\(\,\,\,\,\,f'(x) = \cos x + \sin x\)

 

\(\textbf{5)}\) \( f(x)=5 \sin x+3x \)

Show Answer

The derivative is \( f'(x)=5 \cos x+3 \)

Show Work

\(\,\,\,\,\,\text{Differentiate } f(x) = 5 \sin x + 3x \text{ by applying the derivatives of } \sin x \text{ and } x. \)
\(\,\,\,\,\,f'(x) = 5 \cdot \cos x + 3 \cdot 1\)
\(\,\,\,\,\,f'(x) = 5 \cos x + 3\)

 

\(\textbf{6)}\) \( f(x)=4-\displaystyle\frac{1}{4x^{2}} \)

Show Answer

The derivative is \( f'(x)=\displaystyle\frac{1}{2x^{3} } \)

Show Work

\(\,\,\,\,\,\text{Rewrite } f(x) = 4 – \frac{1}{4x^2} \text{ as } f(x) = 4 – \frac{1}{4}x^{-2} \text{ and use the power rule.}\)
\(\,\,\,\,\,f'(x) = 0 + \frac{1}{4} \cdot (-2)x^{-3}\)
\(\,\,\,\,\,f'(x) = \frac{1}{2x^3}\)

 

See Related Pages\(\)

\(\bullet\text{Derivative Calculator }\)
\(\,\,\,\,\,\,\,\,\text{(Symbolab.com)}\)

\(\bullet\text{ Calculus Homepage}\)
\(\,\,\,\,\,\,\,\,\text{All the Best Topics…}\)

\(\bullet\text{ Definition of Derivative}\)
\(\,\,\,\,\,\,\,\, \displaystyle \lim_{\Delta x\to 0} \frac{f(x+ \Delta x)-f(x)}{\Delta x} \)

\(\bullet\text{ Equation of the Tangent Line}\)
\(\,\,\,\,\,\,\,\,f(x)=x^3+3x^2−x \text{ at the point } (2,18)\)

\(\bullet\text{ Derivatives- Constant Rule}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{d}{dx}(c)=0\)

\(\bullet\text{ Derivatives- Power Rule}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{d}{dx}(x^n)=nx^{n-1}\)

\(\bullet\text{ Derivatives- Constant Multiple Rule}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{d}{dx}(cf(x))=cf'(x)\)

\(\bullet\text{ Derivatives- Sum and Difference Rules}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{d}{dx}[f(x) \pm g(x)]=f'(x) \pm g'(x)\)

\(\bullet\text{ Derivatives- Sin and Cos}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{d}{dx}sin(x)=cos(x)\)

\(\bullet\text{ Derivatives- Product Rule}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{d}{dx}[f(x) \cdot g(x)]=f(x) \cdot g'(x)+f'(x) \cdot g(x)\)

\(\bullet\text{ Derivatives- Quotient Rule}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{d}{dx}\left[\displaystyle\frac{f(x)}{g(x)}\right]=\displaystyle\frac{g(x) \cdot f'(x)-f(x) \cdot g'(x)}{[g(x)]^2}\)

\(\bullet\text{ Derivatives- Chain Rule}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{d}{dx}[f(g(x))]= f'(g(x)) \cdot g'(x)\)

\(\bullet\text{ Derivatives- ln(x)}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{d}{dx}[ln(x)]= \displaystyle \frac{1}{x}\)

\(\bullet\text{ Implicit Differentiation}\)
\(\,\,\,\,\,\,\,\,\)

\(\bullet\text{ Horizontal Tangent Line}\)
\(\,\,\,\,\,\,\,\,\)

\(\bullet\text{ Mean Value Theorem}\)
\(\,\,\,\,\,\,\,\,\)

\(\bullet\text{ Related Rates}\)
\(\,\,\,\,\,\,\,\,\)

\(\bullet\text{ Increasing and Decreasing Intervals}\)
\(\,\,\,\,\,\,\,\,\)

\(\bullet\text{ Intervals of concave up and down}\)
\(\,\,\,\,\,\,\,\,\)

\(\bullet\text{ Inflection Points}\)
\(\,\,\,\,\,\,\,\,\)

\(\bullet\text{ Graph of f(x), f'(x) and f”(x)}\)
\(\,\,\,\,\,\,\,\,\)

\(\bullet\text{ Newton’s Method}\)
\(\,\,\,\,\,\,\,\,x_{n+1}=x_n – \displaystyle \frac{f(x_n)}{f'(x_n)}\)

 

In Summary

The sum and difference rules of derivatives refer to a set of mathematical principles used to calculate the derivative of a function that is the sum or difference of two other functions. These rules allow us to find the derivative of a complex function by breaking it down into simpler parts and using the basic rules of calculus to find the derivative of each component. We learn about derivatives because they are an important tool in the study of calculus and are widely used in fields such as engineering, physics, and economics. Derivatives typically taught in Calculus I.

Related topics to derivatives sum and difference rules include other rules for finding derivatives, such as the product rule, quotient rule, and the chain rule.

Topics that use Derivatives

Optimization: Derivatives can be used to find the maximum or minimum values of a function, which is known as optimization. For example, a company might want to find the production level that maximizes its profits, or an airplane designer might want to find the wing shape that minimizes drag.

Modeling: Derivatives can be used to model real-world phenomena, such as the spread of a disease or the movement of a stock price. By understanding how these quantities change over time, we can make predictions about their future behavior.

Control theory: Derivatives are used in control theory to design control systems for processes or systems that need to be controlled. These might include airplane autopilots, traffic control systems, and manufacturing processes.

Physics: Derivatives are used extensively in physics to describe the motion of objects and the behavior of physical systems. For example, the derivative of a position function gives the velocity of an object.